Simplify and expand the following expression: $ \dfrac{5x}{5x + 9}+\dfrac{5x}{x + 4} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5x + 9)(x + 4)$ Multiply the first term by $\dfrac{x + 4}{x + 4}$ $ \begin{align*} \dfrac{5x}{5x + 9} \times \dfrac{x + 4}{x + 4} & = \dfrac{(5x)(x + 4)}{(5x + 9)(x + 4)} \\ & = \dfrac{5x^2 + 20x}{(5x + 9)(x + 4)}\end{align*} $ Multiply the second term by $\dfrac{5x + 9}{5x + 9}$ $ \begin{align*} \dfrac{5x}{x + 4} \times \dfrac{5x + 9}{5x + 9} & = \dfrac{(5x)(5x + 9)}{(x + 4)(5x + 9)} \\ & = \dfrac{25x^2 + 45x}{(x + 4)(5x + 9)}\end{align*} $ Now we have: $ = \dfrac{5x^2 + 20x}{(5x + 9)(x + 4)} + \dfrac{25x^2 + 45x}{(x + 4)(5x + 9)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5x^2 + 20x + 25x^2 + 45x}{(5x + 9)(x + 4)} $ $ = \dfrac{30x^2 + 65x}{(5x + 9)(x + 4)}$ Expand the denominator: $ = \dfrac{30x^2 + 65x}{5x^2 + 29x + 36}$